Topological censorship and higher genus black holes

Abstract

Motivated by recent interest in black holes whose asymptotic geometry approaches that of anti-de Sitter spacetime, we give a proof of topological censorship applicable to spacetimes with such asymptotic behavior. Employing a useful rephrasing of topological censorship as a property of homotopies of arbitrary loops, we then explore the consequences of topological censorship for the horizon topology of black holes. We find that the genera of horizons are controled by the genus of the space at infinity. Our results make it clear that there is no conflict between topological censorship and the nonspherical horizon topologies of locally anti-de Sitter black holes. More specifically, let D be the domain of outer communications of a boundary at infinity ''scri.'' We show that the principle of topological censorship ͑PTC͒, which is that every causal curve in D having end points on scri can be deformed to scri, holds under reasonable conditions for timelike scri, as it is known to do for a simply connected null scri. We then show that the PTC implies that the fundamental group of scri maps, via inclusion, onto the fundamental group of D: i.e., every loop in D is homotopic to a loop in scri. We use this to determine the integral homology of preferred spacelike hypersurfaces ͑Cauchy surfaces or analogues thereof͒ in the domain of outer communications of any four-dimensional spacetime obeying the PTC. From this, we establish that the sum of the genera of the cross sections in which such a hypersurface meets black hole horizons is bounded above by the genus of the cut of infinity defined by the hypersurface. Our results generalize familiar theorems valid for asymptotically flat spacetimes requiring simple connectivity of the domain of outer communications and spherical topology for stationary and evolving black holes.

Black rings in (Anti)-de Sitter space

“…However, it may still be that fat black rings of larger size exist in deSitter, and they might even reach the size of the cosmological horizon. The existence of black rings in four-dimensional deSitter space is more difficult to rule out than in asymptotically flat or Anti-deSitter spacetimes, where they are forbidden by topological censorship theorems [29]. It is clear that with our methods we can not construct them, since the requisite vacuum four-dimensional black strings do not exist.…”

On the Geometry and Mass of Static, Asymptotically AdS Spacetimes, and the Uniqueness of the AdS Soliton

“…Thus, we can invoke the splitting theorem which tells us that there are k noncompact directions in the universal covering space Σ * , so that Σ * = R k × W , where W is compact. By results on topological censorship [15], the homomorphism i * : π 1 (∂ Σ) → π 1 ( Σ) is onto. But ∂ Σ = A × B, where A is the (n − 2) torus and B is the circle of assumption (a), whence π 1 (∂ Σ) ≈ π 1 (A)×π 1 (B).…”

“…This means that the submanifold W is 2-dimensional. Using the topological censorship theorem [15], we then show that Σ ≃ T n−2 × W . The only undetermined functions are then the 2-dimensional metricσ AB ( y) on W and the lapse N ( y), which, with the aid of the field equations, can be solved for explicitly.…”

“…When this topological restriction is relaxed, black hole spacetimes which are asymptotically locally AdS and which have nonspherical horizons are known to exist 1 (for a review, see [31]). The timelike conformal infinities of these black holes have nonspherical cross-sections [15]. However, a conformal boundary with cross-sections of nonspherical topology cannot also serve as the conformal boundary of a nonsingular locally AdS spacetime.…”

“…(Hence, each segment of a null line is maximal with respect to the Lorentzian distance function.) Arguments involving null lines have arisen in numerous situations, such as the Hawking-Penrose singularity theorems [20], results on topological censorship ( [15], and references cited therein), and the Penrose-Sorkin-Woolgar approach to the positive mass theorem [33,41] and related results on gravitational time delay [18]. It will be convenient for the purposes of the present paper to require null lines to be not only inextendible, but geodesically complete.…”